Wednesday, May 23rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Benson Au (Berkeley)

Title: Rigid structures in the universal enveloping traffic space

Abstract:  For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, Cébron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that extends the trace $\varphi$. The CDM construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure.

We show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a duality between classical independence and free independence.

We apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of Bryc, Dembo, and Jiang and of Mingo and Popa. This is joint work with Camille Male.

Free Probability and Random Matrices Seminar Webpage:

Tuesday, April 3rd, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Pei-Lun Tseng (Queen's University)

Title: Linearization trick of infinitesimal freeness II

Abstract:  Last week, we introduced how to find a linearization for a given selfadjoint polynomial and showed some properties of this linearization. We will continue our discussion this week and introduce the operator-valued Cauchy transform. Then, we will show the algorithm for finding the distribution of $P$ where $P$ is a selfadjoint polynomial with selfadjoint variables $X$ and $Y$. Based on this method, we will discuss how to extend this algorithm for finding infinitesimal distribution for $P$.

Free Probability and Random Matrices Seminar Webpage:

Tuesday, March 20th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Mihai Popa (University of Texas, San Antonio)

Title: Permutations of Entries and Asymptotic Free Independence for Gaussian Random Matrices

Abstract:  Since the 1980's, various classes of random matrices with independent entries were used to approximate free independent random variables. But asymptotic freeness of random matrices can occur without independence of entries: in 2012, in a joint work with James Mingo, we showed the (then) surprising result that unitarily invariant random matrices are asymptotically (second order) free from their transpose. And, in a more recent work, we showed that Wishart random matrices are asymptotically free from some of their partial transposes. The lecture will present a development concerning Gaussian random matrices. More precisely, it will describe a rather large class of permutations of entries that induces asymptotic freeness, suggesting that the results mentioned above are particular cases of a more general theory.

Free Probability and Random Matrices Seminar Webpage:

Tuesday, March 6th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Jamie Mingo (Queen's University)

Title: The Infinitesimal Law of a real Wishart Matrix

Abstract:  The Wishart ensemble is the random matrix ensemble used to estimate the covariance matrix of a random vector. Infinitesimal freeness is a generalized independence stronger than freeness but weaker than second order freeness. I will give the infinitesimal distribution of a real Wishart matrix. It is given in terms of planar diagrams which are 'half' of a non-crossing annular partition.

Free Probability and Random Matrices Seminar Webpage:

Tuesday, February 6th, 2018

Time: 3:30-5:00 p.m.  Place: Jeffery Hall 319

Speaker: Jamie Mingo (Queen's University)

Title: The Infinitesimal Law of the GOE

Abstract:  If $X_N$ is the $N \times N$ Gaussian Orthogonal Enemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

Free Probability and Random Matrices Seminar Webpage: